×

zbMATH — the first resource for mathematics

One-dependent trigonometric determinantal processes are two-block-factors. (English) Zbl 1067.60010
If \(f\) is a trigonometric polynomial of degree \(m\), a stationary process via determinants of the Toeplitz matrix for \(f\) can be defined. This process is \(m\)-dependent. An \((m+1)\)-block-factor is trivially \(m\)-dependent. The problem is to see if all \(m\)-dependent processes are \((m+1)\)-block-factors. A positive answer is given for \(m=1\), namely: one-dependent trigonometric determinantal processes are two-block-factors.

MSC:
60G10 Stationary stochastic processes
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Aaronson, J., Gilat, D. and Keane, M. (1992). On the structure of 1-dependent Markov chains J. Theoret. Probab. 5 545–561. · Zbl 0754.60070
[2] Aaronson, J., Gilat, D., Keane, M. and de Valk, V. (1989). An algebraic construction of a class of one-dependent processes. Ann. Probab. 17 128–143. JSTOR: · Zbl 0681.60038
[3] Burton, R. M., Goulet, M. and Meester, R. (1993). On 1-dependent processes and \(k\)-block factors. Ann. Probab. 21 2157–2168. JSTOR: · Zbl 0788.60049
[4] Götze, F. and Hipp, C. (1989). Asymptotic expansions for potential functions of i.i.d. random fields. Probab. Theory Related Fields 82 349–370. · Zbl 0659.60035
[5] Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables . Wolters-Noordhoff, Groningen. · Zbl 0219.60027
[6] Janson, S. (1983). Renewal theory for \(m\)-dependent variables. Ann. Probab. 11 558–568. JSTOR: · Zbl 0514.60086
[7] Janson, S. (1984). Runs in \(m\)-dependent sequences. Ann. Probab. 12 805–818. JSTOR: · Zbl 0545.60080
[8] Lyons, R. (2003). Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98 167–212. · Zbl 1055.60003
[9] Lyons, R. and Steif, J. (2003). Stationary determinantal processes: Phase transitions, Bernoullicity, entropy, and domination. Duke Math. J. 120 515–575. · Zbl 1068.82010
[10] Soshnikov, A. (2000). Determinantal random point fields. Uspekhi Mat. Nauk 55 107–160. · Zbl 0991.60038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.